Sep 24, 2013 - We find a simple analytic description of this process and show that while the ... Based on this, the COPE pulse is related to Hawking radiation, ...
OPEN SUBJECT AREAS: ULTRAFAST LASERS PHOTONIC CRYSTALS NONLINEAR OPTICS
Cavity Optical Pulse Extraction: ultra-short pulse generation as seeded Hawking radiation Falk Eilenberger1, Irina V. Kabakova2, C. Martijn de Sterke2, Benjamin J. Eggleton2 & Thomas Pertsch1
HIGH-ENERGY ASTROPHYSICS 1
Received 13 February 2013 Accepted 5 August 2013 Published 24 September 2013
Correspondence and requests for materials should be addressed to F.E. (falk.eilenberger@ uni-jena.de)
Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-Universita¨t Jena, Max-Wien-Platz 1, 07743 Jena, Germany, 2ARC Center for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Institute of Photonics and Optical Sciences (IPOS), School of Physics, University of Sydney, New South Wales, 2006, Australia.
We show that light trapped in an optical cavity can be extracted from that cavity in an ultrashort burst by means of a trigger pulse. We find a simple analytic description of this process and show that while the extracted pulse inherits its pulse length from that of the trigger pulse, its wavelength can be completely different. Cavity Optical Pulse Extraction is thus well suited for the development of ultrashort laser sources in new wavelength ranges. We discuss similarities between this process and the generation of Hawking radiation at the optical analogue of an event horizon with extremely high Hawking temperature. Our analytic predictions are confirmed by thorough numerical simulations.
L
asers1,2 require an energetically charged (‘‘inverted’’) medium, placed inside a cavity. The medium usually consists of excited molecules that interact coherently with a standing wave formed inside the resonator by means of stimulated emission, generating coherent light when the molecules transit to a lower energy state. As a consequence of the discrete nature of the transition process, each host molecule emits only a limited number of wavelengths, restricting the possibility to generate ultrashort laser pulses, which are intrinsically broadband. This restriction is linked to the dual purpose of the lasing material, which acts as an energy storage and also provides a transition with the desired photon energy. Thus ultrashort lasers are limited to a few lasing media and wavelength ranges, mostly in the visible or near-infrared. In this paper we present Cavity Optical Pulse Extraction (COPE), which readily lends itself as a novel approach for the generation of ultrashort, arbitrary wavelength pulses. The key element is that the storage of energy and the generation of ultrashort pulses are separated, which allows us to use any cw-transition as a master oscillator for an ultrashort pulse, shaped by an ultrashort trigger of a different wavelength. A schematic of the proposed setup is shown in Fig. 1, which underlines its conceptual simplicity. It is based on the established concept of the optical pushbroom3–5; where cross-phase modulation induced by a strong trigger pulse ‘‘pushes’’ a slowly propagating burst of light out of a fiber Bragg grating (FBG). In contrast to a conventional push broom scheme, we exploit the intensity enhancement of the resonant cavity mode being extracted by the trigger pulse. We apply the idea to an all-fiber geometry in which the cavity is formed by an optical fiber Bragg grating6,7, employing dynamic wavelength conversion8,9 in the ultrafast regime. In contrast to previous works on pulse generation using dynamically tuned cavities in the adiabatic regime where the transition time is longer than the roundtrip time10–12, we exploit an ultrafast non-adiabatic process acting like a shock-front propagating through the cavity13. As a consequence, a short pulse with nanojoule energy or more is extracted from the cavity, even if it has only a moderate Q-factor. Unlike traditional cavity dumping schemes14, COPE does not require a short pulse to be in the cavity but generates it during the extraction process. This results in a light source, which combines the wavelength of an external pump source, with the power enhancement of a grating cavity, and the pulse width of an ultrashort oscillator. We argue that COPE can be considered to be a classical analogue of Hawking radiation15–18 – generating new wavelengths by means of resonant generation of radiation19–21 close to an optical event horizon22–25. The event horizon is here represented by a refractive index perturbation, induced by cross-phase modulation, moving at the speed of light through the material. Based on this, the COPE pulse is related to Hawking radiation, accumulating close to a moving index perturbation. The outline of this paper is as follows. We first present a simple analytic model of the COPE process, and then in a separate section point out and discuss the analogy to Hawking radiation and event horizons. We then compare
SCIENTIFIC REPORTS | 3 : 2607 | DOI: 10.1038/srep02607
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Figure 1 | Schematic of the COPE setup. A cw-laser (green wave) excites a cavity mode (green arch) in a FBG cavity (striped cylinder). A counter propagating TP (red pulse) interacts with the energy stored in the cavity and extracts a short pulse, with the same wavelength as the cavity mode (green pulse, superimposed on red pulse). (Inset) Refractive index modulation (blue) and modal profile at the Bragg wavelength (red) of three fiber Bragg gratings (FBGs). (a) Uniform FBG. (b) Uniform FBG cavity with p-phase shift in the center. (c) Sinusoidal FBG cavity as discussed in this paper.
our result with fully numerical calculations, and discuss issues related to experimental observation. We then briefly discuss our results and conclude.
Results Analytic description of the COPE process. The propagation of light in a shallow Bragg grating26 is described by the set of coupled mode equations27,28, where for convenience we measure all times in units of c/n, LA+ LA+ zi zdA+ zkðz ÞA+ z2cPðz,t ÞA+ ~0 +i Lz Lt
ð1Þ
for A6(z,t), the amplitudes of the mode propagating in forward (1) and backward (2) directions. The frequency detuning is the deviation of the inverse wavelength l21 d~2p l{1 {l{1 B from the Bragg resonance l{1 of the grating, which depends on B the grating period L as lB 5 2nL. The Bragg wavelength must match a CW laser source but is otherwise arbitrary. The grating itself is characterized by its coupling strength k(z) 5 pDn(z)/lB, proportional to the local, periodic refractive index modulation Dn(z). Depending on the shape of k(z) an FBG can work as a Bragg reflector or exhibit resonances similar to a Fabry-Perot resonator with distributed feedback. Some examples are given in the inset of Fig. 1. Here we consider a FBG cavity with k(z) 5 k0 sin(p L21 z) and of length 2L (see Fig. 1(c)), which has a cavity mode at a detuning of d 5 0 with a simple Gaussian form (see Appendix), which allows us to obtain analytic solutions. In a practical implementation of COPE, the specific shape of the cavity mode does not matter, and the better-known cavities consisting of a uniform grating with a p-phaseshift6,29 can be used instead (see Appendix). The cavity needs to be sufficiently strong such that k0 L?1:
ð2Þ
The trigger pulse (TP) interacts with the COPE field via cross phase modulation with the TP’s intensity P(z,t); we thus model the TP as an invariant, moving refractive index perturbation. The interaction strength is determined by the cross-phase modulation constant c SCIENTIFIC REPORTS | 3 : 2607 | DOI: 10.1038/srep02607
5 2pv/(cAeff), where Aeff is the effective area for the nonlinear interaction between the cavity field and the TP30. Our model aims to provide a fundamental understanding of the COPE process and makes various approximations to achieve this. First we ignore dispersive and/or nonlinear reshaping of the TP during propagation; we do so since the cavity length is usually a few millimeters and therefore much shorter than common dispersion lengths. Furthermore, nonlinear reshaping of the pulse intensity can be slow if the TP experiences normal dispersion, such that soliton self-compression, etc. are absent. The same is true for the COPE pulse. Other approximations, in particular those related to group velocity and dispersion are elaborated in the discussion. We now show how a substantial fraction of the cavity’s energy can be extracted in a coherent ultrashort burst, through cross-phase modulation by the TP. For simplicity we take a TP with a Gaussian shape with an FWHM t traveling through the grating in the -z direction, such that ðzzt{t0 Þ2 Pðz,t Þ~P0 exp {4 lnð2Þ : ð3Þ t2 It is convenient to decompose the cavity field into a component ð0Þ trapped in the cavity mode A+ ðz Þ, excited by a cw-laser tuned to the cavity resonance and whose shape is given by Eq. (A.17), and the extracted COPE pulse with field B6, which forms in the interð0Þ action with the TP. Writing A+ ðz,t Þ~A+ ðz ÞzB+ ðz,t Þ, and using Eq. (A.17), we find p LBz LBz zi zk0 sin z B{ z2Pðzzt{t0 Þ zi Lz Lt L ð0Þ A ðz ÞzBz ~0 ð4Þ p LB{ LB{ {i zi zk0 sin z Bz z2Pðzzt{t0 Þ Lz Lt L iAð0Þ ðz ÞzB{ ~0: The spatial Fourier transform of these equations can be written as {i
LB+ k0 LB+ ~+bB+ {ip Lt L Lb z½expð{ibðt{t0 ÞÞPðbÞ6B+
ð5Þ
zv+ ½expð{ibðt{t0 ÞÞPðbÞ6Að0Þ ðbÞ, where A(b)flB(b)denotes convolution, n1 5 1 and n2 5 i. We now introduce a second approximation, which allows us to simplify Eq. (5) further. We assume that the pump pulse width t is much shorter than the localization length of the grating k{1 0 such that t=k{1 0 =L:
ð6Þ
This allows us to ignore the second term on the right hand side of Eq. (5) in favor of the first. With this approximationqwe replace the ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
parabolic dispersion relation of the grating v~+ k20 zb2 31,32, by a light-cone like dispersion relation v 5 6 b. This is justified because the spectrum of the extracted pulse is predominantly at frequencies jdj?k0 , where the difference between these is small. We discuss this further below. Next we simplify the last term in Eq. (5), corresponding to the TP. We use the approximations of Eqs. (2) and (6) and the explicit expression for the trapped field Eq. (A.17) derived in the Appendix, and Eq. (3) for the definition of the TP, to find ½expð{ibðt{t0 ÞÞPðbÞ6Að0Þ ðbÞ < expð{ibðt{t0 ÞÞPðbÞAð0Þ ðt{t0 Þ,
ð7Þ 2
www.nature.com/scientificreports i.e., the interaction between the trapped field and the TP is concentrated at the TP’s center. We now transform the COPE pulse into a frame of reference co^ + expð+ibðt{t0 ÞÞ and apply moving with the TP, such that B+ ~B Eq. (6) and Eq. (7) to the evolution equation, resulting in: ^z LB ^ z z expð{2ibðt{t0 ÞÞ {i ~½expð{2ibðt{t0 ÞÞPðbÞ6B Lt PðbÞAð0Þ ðt{t0 Þ {i
ð8Þ
^{ LB ^ { ziPðbÞAð0Þ ðt{t0 Þ: ~PðbÞ6B Lt
The transformation to this frame has a clear physical meaning: it is the frame in which the TP is stationary and the frequency of a wave d9 is conserved. This not true in the laboratory rest frame, where new frequencies d are generated because the TP moves. Transformation between the two frames is accommodated by a Doppler frequency shift d9 5 d 1 b. We now derive, starting from Eqs. (8), an analytic expression for the development of the COPE pulse. The TP drives the generation of the COPE pulse via the interaction with the trapped field, which is described by the second term on the right hand side of Eq. (8). The ^ + are, at this level of approximation, evolution of the COPE fields B ^ + lie outdecoupled, because most of the spectral components of B side the grating-induced band-gap. We can therefore treat the two ^ z
